text: Luke's Lagrangian formulation is for non-linear surface gravity waves on an—incompressible, irrotational and inviscid—potential flow.
  The relevant ingredients, needed in order to describe this flow, are:
  * is the velocity potential,
  * is the fluid density,
  * is the acceleration by the Earth's gravity,
  * is the horizontal coordinate vector with components  and ,
  * and  are the horizontal coordinates,
  * is the vertical coordinate,
  * is time, and
  * is the horizontal gradient operator, so  is the horizontal flow velocity consisting of  and ,
  * is the time-dependent fluid domain with free surface.
  The Lagrangian , as given by Luke, is:
  From Bernoulli's principle, this Lagrangian can be seen to be the integral of the fluid pressure over the whole time-dependent fluid domain . This is in agreement with the variational principles for inviscid flow without a free surface, found by Harry Bateman.
  Variation with respect to the velocity potential  and free-moving surfaces like  results in the Laplace equation for the potential in the fluid interior and all required boundary conditions: kinematic boundary conditions on all fluid boundaries and dynamic boundary conditions on free surfaces. This may also include moving wavemaker walls and ship motion.
  For the case of a horizontally unbounded domain with the free fluid surface at  and a fixed bed at , Luke's variational principle results in the Lagrangian:
  The bed-level term proportional to  in the potential energy has been neglected, since it is a constant and does not contribute in the variations. 
  Below, Luke's variational principle is used to arrive at the flow equations for non-linear surface gravity waves on a potential flow.